Preference Relations as the Information

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represented in one of the following three ways: 1. A preference ordering of the alternatives. In this case, an expert, ek, provides his preferences on. X as anĀ ...
Preference Relations as the Information Representation Base in Multi-Person Decision Making F. Chiclana

F. Herrera

E. Herrera-Viedma

Dept. of Computer Science and Arti cial Intelligence E.T.S. de Ingeniera Informatica University of Granada, 18071- Granada, Spain herrera,[email protected]

Abstract In a fuzzy multi-person decision making problem, where each expert provides information about the alternatives in di erent ways, several techniques to make the information representation uniform are given. Assuming that the experts may provide their opinions by means of preference orderings, or utility functions or preference relations, the fuzzy preference relations are chosen as usual representation element and then, from them any election process may be developed.

1 INTRODUCTION As is well known, fuzzy theories and methodologies are very useful to deal with imprecise and vague information in many problems. In particular, in decision making problems where human judgments including preferences often vague. Some issues about this topic are presented in [3, 8, 9, 10, 11]. The application of Fuzzy Set Theory in real world decision making problems has given very good results. Its aim feature is that it provides a more exible framework, where it is possible to simulate human's abilities to deal with the fuzziness of human judments quantitatively, and therefore to incorporate more human consistency in decision making models. Usually, nonfuzzy preferences may be represented as the set of preferred alternatives (choice set), preference relations (orderings), or utility functions (cardinal) [16]. Analogously, we may consider the following three representations of fuzzy preferences [16]: fuzzy choice sets, fuzzy preference relations, and fuzzy utility functions. Some details of these fuzzy representations

and some relationships between nonfuzzy and fuzzy preferences are discussed in [16]. Multi-Person Decision Making (MPDM), which occurs whenever two or more people actively participate in arriving at decision, is a ubiquitous human activity. Many formal approaches to multiperson decision mak-

ing have been developed over the years and numerous mathematical models have been proposed in their analysis [4, 5, 6, 7, 9]. In a usual model of the MPDM problem with fuzzy preference relations is assumed that there exists a nite set of alternatives X = fx1; x2; : : :; xng; (n  2) as well as a nite set of experts E = fe1 ; e2; : : :; em g; (m  2). Each expert, ek 2 E, provides his preference on X by means of a fuzzy preference relation, P k  XxX, with membershipk function, P : XxX ! [0; 1]; where P (xi ; xj ) = pij denotes the preference degree of the alternative xi over xj . With a view to build a more exible framework and to give more freedom degree to the experts in the representation of their preferences, we can consider a MPDM model in which the experts may provide their preferences in any of these three ways:  As a preference ordering of the alternatives. In this case the alternatives are ordered from the best to the worst, without any other supplementary information.  As a fuzzy preference relation. This is the usual case, i.e., when an expert supplies a fuzzy binary relation over the set of alternatives, re ecting the degree to which an alternative is prefered to another.  As an utility function. In this case an expert supplies an real evaluation (physical or monetary value) for each alternative, i.e., a function that associates each alternative with a real number indicating the performance of that alternative according to his point of view. Assuming this MPDM problem, where each expert supplies his preferences about the alternatives in different way, our proposal is to establish an uniform representation of the preferences that allows us to develop a more general MPDM model. We propose to use the fuzzy preference relation as the main element of the uniform representation of the preferences. Therefore, we propose to transform the di erent representations of the preferences to fuzzy preference relations. This solution may be a good one becuase the use of fuzzy k

k

preference relations in decision making situations appears to be a useful tool in modelling decision processes, overcoat when we want to aggregate experts' preferences into group preferences, that is, in the resolution processes of the MPDM problems [4, 5, 6, 7, 15]. Furthermore, preference orderings and utility values are included in the family of fuzzy preference relations [16] and most of the existing results on MPDM under fuzzy preferences are dealing with fuzzy preferences relations [4, 5, 6, 7, 8, 11, 15]. To do so, in the following section is presented the framework of the MPDM problem considered. Section 3 shows di erent transformation functions of preference orderings and utility values to achieve a uniform representation based on fuzzy preference relations. Section 4 presents a concrete example of application of some transformation functions in a MPDM problem with nonuniform preferences, using a selection process based on the OWA (Ordered Weighted Averaging) operator [17] and on the quanti er guided nondominance degree [1]. Finally, some conclusions are pointed out.

In this context, the resolution process of the MPDM problem consists of obtaining a set of solution alternatives, Xsol  X; from the preferences given by the experts. Since the experts provide their preferences in di erent ways, to obtain a uniform representation of the preferences must be the rst step of the resolution process of the MPDM problem. Achieved this uniform representation, we can develop from it any known selection processes [6, 1]. In this sense, the resolution process of the MPDM problem considered presents the scheme given in the Figure 1. In the next EXPERT SET

PREFERENCES

-PREFERENCE ORDERINGS -UTILITY FUNCTIONS -FUZZY PREFERENCE RELATIONS

2 PRELIMINARIES The particular MPDM problem that we consider presents the following setting. We have a nite set of alternatives, X = fx1 ; x2; : : :; xn: (n  2)g; to be analyzed by a nite set of experts, E = fe1 ; e2; : : :; em : (m  2)g. As each expert is characterized by his own ideas, attitudes, motivations and personality, it is quite natural to think that di erent experts will provide their preferences in a di erent way. Then, we assume that the experts' preferences may be represented in one of the following three ways: 1. A preference ordering of the alternatives. In this case, an expert, ek , provides his preferencesK on X as an individual preference ordering, O = fok (1); :::; ok(n)g, where ok () is a permutation function over the index set f1; :::; ng for the expert ek [1, 14]. Therefore, according to the view point of each expert, an ordered vector of alternatives, from the best one to the worst one, is given. 2. A fuzzy preference relation. With this representation, the expert's preferences on X is described by a fuzzy binary relation, P k  XxX, with membership function, P : XxX ! [0; 1]; where pkij denotes the preference degree of the alternative xi over xj [6, 8, 11, 15]. We assume that P k is reciprocal without loss of generality, i.e., by de nition, (i) pkij + pkji = 1 and (ii) pkii = 0; 8 i; j; k. 3. An utility function. In this case, an expert, ek , provides his preferences on X as a set of n utility values, U k = f; uki; i = 1; :::; ng; uki 2 [0; 1], where uki represents the utility evaluation given by the expert ek to the alternative xi [12, 16]. k

TRANSFORMATION FUNCTIONS UNIFORM REPRESENTATION BASED ON FUZZY PREFERENCE RELATIONS

SELECTION PROCESS SET OF SOLUTION ALTERNATIVES

Figure 1: Resolution Process of the MPDM Problem. section we study the problem of the uniform representation and analyze di erent transformation functions to achieve an uniform representation, which is based on fuzzy preference relations.

3 AN UNIFORM REPRESENTATION BASED ON FUZZY PREFERENCE RELATIONS As we said at the beginning, we propose to use fuzzy preference relations as the base element of the uniform representation. Therefore, as it is showed in Figure 1, we need some transformation functions to transform preference orderings and utility values into fuzzy preference relations. In the next subsections we analyze this topic.

3.1 TRANSFORMING PREFERENCE ORDERINGS INTO FUZZY PREFERENCE RELATIONS Let's assume that each expert, ek , provides his preferences on X by means of a preference ordering, Ok = fok (1); :::; ok(n)g: For every preference ordering, Ok , we will suppose, whithout loss of generality, that the lower the position of an alternative in a preference ordering, implies the better the alternative satis es the expert. For example, suppose that an expert , ek , supplies his preferences about a set of four alternatives, X = fx1; x2; x3; x4gk; by means of the following ordering preference, O = f3; 1; 4; 2g. In this case ok (1) = 3, ok (2) = 1, ok (3) = 4; ok (4) = 2. This means that alternative x2 is the best for that expert, while alternative x3 is the worst. We proposed in [1] a rst approach to derive a fuzzy preference relation from a preference ordering. Clearly, an alternative sati es an expert more or less depending on its position in his preference ordering, i.e., the lower the position of an alternative in the preference ordering, the better the alternative satis es the expert, and viceversa. Therefore, in our approach, we considered that for an expert, ek , his preference value of the alternative xi over xj , pkij , depends only on the values of ok (i) and ok (j), i.e., we assert that there exists a transformation function, f, that assigns a credibility value of preference of any alternative over any other alternative, from any preference ordering, pkij = f(ok (i); ok (j)): This transformation function, f, must satisfy that the more ok (i) the less pkij , and the more ok (j) the more is pkij . Therefore, it must be a non increasing function of the rst argument and a non decreasing function of the second argument. An example of this type of transformation functions are those that obtain the credibility value of preference of any alternative over any other alternative depending on the value of the di erence between the alternatives' positions, i.e., pkij = g(ok (j) ? ok (i)); where g is a non decreasing function. For example, in [1] we use the following transformation function:  k > ok (i) k 1 k k pij = g (o (j) ? o (i)) = 10 ifif ook (j) (i) > ok (j) This transformation function, g1 , derives non-fuzzy preference relations, where pkij re ects the degree in f0; 1g to which xi is declared not worse than xj for the expert, ek . In our example, the alternative x2 is not worse than alternatives x4, x1 , x3; the alternative x4 is not worse than alternatives x1 , x3; and, nally, the alternative x1 is not worse than alternative x3. Therefore, we obtain the following non-fuzzy prefer-

ence relation:

2? 0 1 0 3 1 1 75 P k = 64 01 0? ? 0 1 0 1 ? The simplicity and easy use are, in our opinion, the only virtues of this particular transformation function, g1 . This function does not re ect any kind of intensity of preference of an alternative over another when we compare pairs of alternatives, that is, for example, it does not distinguish between the preference of alternative x2 over x4 and the preference of alternative x2 over x3 . Therefore, in these situations we need to use another type of function which re ects appropriately the di erent positions between alternatives, and for example, if pk24 = 2=3 then pk41 and pk13 should be equal to 2=3, but pk21 should be greater than or equal 2=3 and less than or equal pk23: A posibility consists of giving a value of importance or utility to each alternative in a way such that the lower the position of an alternative, the higher the value of utility. In this case, we can give those values as a diference scale. We can assing the following value, vik , k vik = 1 ? o n(i)??1 1 ; as a degree of utility of the kalternative xi according to the preference ordering, O , provided by an expert, ek . It is clear that the set of n utility values obtained is normalized, that is, MAXi fvik g ? MINi fvik g  1: In this context, we can use the following expression to obtain the preference values, pkij , pkij = 21 (1 + vik ? vjk ); i.e., k k pkij = 21 (1 + no ?(j)1 ? no ?(i)1 ); and therefore, the preference value is given by means of a transformation function, g2 , depending on the difference ok (j) ? ok (i), pkij = g2 (ok (j) ? ok (i)): In our example, using g2 we obtain the following fuzzy preference relation: 2 ? 1=6 4=6 2=6 3 5=6 ? 1 4=6 75 P k = 64 2=6 0 ? 1=6 4=6 2=6 5=6 ? Both transformation functions, g1 and g2 ; allow preference values, pkij ; to verify these two relationships:

 0  pkij  1; 8i; j  pkij + pkji = 1; 8i; j:

Furthermore, when there is indi erence between two alternatives, that is, when ok (i) = ok (j), g2 gives a value of pkij equals to 1=2 and g1 can be easily modi ed to verify this property. The transformation function, g2 , has been investigated by Dombi [2], and proposed by Tanino in [15]. Dombi de ned an universal preference function, being g2 a particular case of that function, and showed that the utility based decision making gives the same result as the preference based using the universal preference function. The relationship between preference orderings, utility values given on the basis of a di erence scale and preference relations is a problem to be discussed in future works.

3.2 TRANSFORMING UTILITY VALUES INTO FUZZY PREFERENCE RELATIONS In this case, we assume that each expert, ek , provides his preferences on X by means of a set of utility values, U k = fuki ; i = 1; :::; ng: Here, each alternative, xi , is supposed to have associated a real number, uki ; indicating the performance of that alternative accordingk to the expert ek . For every set of utility values, U , we will suppose, whithout loss of generality, that the higher the evaluation, the better the alternative satis es the expert. Any possible transformation function, h, to derive a fuzzy preference relation from a set of utility values, must obtain, for an expert, ek , his preference value of the alternative xi over xj , pkij , depending only on the values of uki and ukj , i.e., pkij = h(uki ; ukj ): This transformation function, h, must satisfy that the more uki the more pkij , and the more is ukj the less is pkij . Therefore, it must be a non decreasing function of the rst argument and a non increasing function of the second argument. An example of this type of transformation functions are those that obtain the credibility value of preference of any alternative over any other alternative depending on the value of the quotient between the respective utility values of the alternatives, i.e., k

pkij = l( uki ); uj where l is a non decreasing function. Then, interpreting uu as a ratio of the preference intensity for xi to that of xj , that is, xi is uu times as good as xj , a possible transformation function to obtain the intensity of preference of the alternative xi over alternative xj for k i k j

k i k j

expert ek , pkij ; is given by uki

k k )2 u i ; i 6= j: pkij = l1 ( uuki ) = u u = (uk )(u 2 + (uk )2 j j i + u u This type of transformation functions, l, have been investigated by Luce and Suppes [12]. Others examples of this type of functions may be found in [12, 15]. The relation between utility values given on the basis of a positive ratio scale and preference relations is a problem to be discussed in future works. In the following section we present an example of a MPDM problem where it is showed the use of the some studied transformation functions. This example is developed using a selection process based on the OWA operator [17] and the quanti er guided non dominance degrees [1]. k j

k i k j

k j k i

4 AN EXAMPLE OF A MPDM PROBLEM WITH NON UNIFORM PREFERENCES Let X = fx1; x2; x3; x4g be a set of four alternatives to be analyzed by a set of four experts E = fe1 ; e2; e3 ; e4g. The experts' preferences are the following:  e1 : O1 = f3; 1; 4; 2g.  e2 : U 2 = f0:5; 0:8;0:4; 0:5g.  e3 : U 3 = f0:3; 0:7;0:4; 0:6g.  e4 : 2 ? 0:2 0:3 0:4 3 ? 1 0:5 7 P 4 = 64 0:8 0:7 0 ? 0:4 5 : 0:6 0:5 0:6 ? Then, the rst step of the resolution process of the MPDM problem consists of achieving a uniform representation. Using the transformation functions g2 and 1 l for preference ordering and sets of utility values, respectively, we obtain the following fuzzy preference relations: 2 ? 0:16 0:67 0:33 3 ? 1 0:67 75 P 1 = 64 0:84 0:33 0 ? 0:16 0:67 0:33 0:84 ? 2 ? 0:28 0:61 0:5 3 0:72 ? 0:8 0:72 75 P 2 = 64 0:39 0:2 ? 0:39 0:5 0:61 ? 2 ? 0:28 0:15 0:36 0:2 3 0:85 ? 0:75 0:58 75 P 3 = 64 0:64 0:25 ? 0:31 0:8 0:42 0:69 ? From the four fuzzy preference relations applying a selection process designed in [1], the set of solution alternatives, Xsol , is achieved. This selection process is developed in the following phases.

4.1 PHASE I: AGGREGATION In this phase from the set of the fuzzy preference relations, fP k ; k = 1; : : :; mg, we will derive a collective fuzzy preference relation, P c , which is a fuzzy preference relation of the group of experts as a whole. This collective relation is obtained using the aggregation operator, the OWA operator [17], Q , guided by fuzzy quanti ers [18], Q, representing the concept of fuzzy majority. The derivation expression of the collective relation, P c, is the following: P c = Q (P 1; :::; P m); where the aggregation operator, Q , and the fuzzy quanti er, Q, are de ned as follows: De nition 1. [17] Let fa1 ; : : :; am g; be a list of values to aggregate, then the OWA operator Q is de ned as Q (a1; : : :; am ) = W  B T = mi=1 wi  bi ; where W = [w1; : : :; wm ], is a weighting vector, such that,

wi = Q(i=m) ? Q((i ? 1)=m); i = 1; : : :; m; and therefore, wi 2 [0; 1] and i wi = 1; and B is the associated ordered value vector. Each element bi 2 B is the i-th largest value in the collection a1; : : :; am :

De nition 2. [18] Q is a non decreasing proportional fuzzy quanti er de ned as a fuzzy subsets of the unit interval, [0,1], in such a way that for any r 2 [0; 1],

Q(r) indicates the degree to which the proportion r is compatible with the meaning of the quanti er it represents. The membership function of a non-decreasing proportional fuzzy quanti er, Q, can be represented as 8 < 0 if r < a Q(r) = : rb??aa if a  r  b a; b; r 2 [0; 1]: 1 if r > b

Some examples of proportional fuzzy quanti ers are:

"Most", "At least half", "As many as possible", with

the parameters (a,b), (0.3,0.8), (0,0.5), (0.5,1), respectively. Therefore, following with our example, we obtain the collective relation, 2? 0:1585 0:33 0:25 3 0:76 ? 0:775 0:54 75 P c = 64 0:345 0 ? 0:235 0:55 0:29 0:605 ? using the OWA operator, Q; with the fuzzy quanti er "As many as possible", i.e., with the weighting vector, W = [0; 0; 0:5; 0:5]:

4.2 PHASE II: EXPLOITATION In this phase, from the collective fuzzy preference relation, P c, applying the choice degree of the alternatives, quanti er guided non dominance degree, de ned in [1], the set of solution alternatives, Xsol , is obtained. This

degree is based on the use of the OWA operators, Q, and generalizes the Orlovski's non dominated alternative concept [13].

De nition 3. [1] The quanti er guided nondominance degree, QGNDDi , of an alternative, xi , is used to quantify the degree to which the alternative, xi , is not dominated by a fuzzy majority of the remaining alternatives, and is de ned according to the following expression: where

QGNDDi = Q (1 ? psji; j = 1; :::; n; j 6= i)

psji = maxfpcji ? pcij ; 0g; represents the degree to which xi is strictly dominated by xj . Then, obtaining the quanti er guided non-dominance degree, QGNDDi , of each alternative, xi , from the collective relation, P c, the set of solution alternatives, Xsol ; is achieved according to the following expression: Xsol = fxi j xi 2 X; QGNDDi = supx 2X QGNDDj g: j

Therefore, following with our example, we obtain this strict fuzzy preference relation, P s , 2? 0 0 0 3 0:5915 ? 0:775 0:25 75 : P s = 64 0:015 0 ? 0 0:3 0 0:37 ? Then, using the OWA operator with the same fuzzy quanti er "As many as possible", but this time, with the weighting vector, W = [0; 0:334;0:666]; we obtain the following alternative quanti er guided nondominance degrees of alternatives: x1 x2 x3 x4 QGNDDi 0:5 1 0:36 0:83: and thus, the set of solution alternatives is this Xsol = fx2g:

5 CONCLUSIONS In this work, assuming a fuzzy MPDM problem, where each expert provides his preferences about the alternatives in di erent ways, preference ordering, fuzzy preference relations and utility functions, several techniques to make the preferences representation uniform are given. We have used the fuzzy preference relation as the main element of the uniform representation of the preferences, and we have presented di erent transformation functions between preference orderings and fuzzy preference relations, and between utility values and fuzzy preference relations. In this way, we have established an uniform representation of the preferences that allows us to develop more general MPDM models, where the experts have more freedom degree to express his preferences.

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